Download document

Stochastic Noise Analysis of Neural Interface Front End
Amir Zjajo, Carlo Galuzzi, Rene van Leuken
Circuits and Systems Group
Delft University of Technology
Mekelweg 4, 2628 CD Delft, The Netherlands
Abstract— A time-domain methodology for noise analysis of
neural interface front-end with arbitrary deterministic neuron
model excitations is presented. Rather than estimating noise
behavior by a population of realizations, the neural interface
front-end is described as a set of stochastic differential equations
and closure approximations are introduced to obtain the noise
variances, covariances and cross-correlations between any
electrical quantity and any stochastic source as a function of
time. Statistical simulation shows that the proposed method offer
an accurate and an efficient solution closely approximating those
from a time-domain Monte Carlo analysis.
I.
INTRODUCTION
The recent trend in brain machine interfaces for neural
electrophysiological recording [1] has largely been motivated
by the growing interest in observing large scale neuronal
activity. Quantifying noise in cellular dynamics and the
physical electronic interface is one of the central challenges in
the heterogeneous neural simulation and neural rehabilitation
[2] including neural prosthetics and closed-loop stimulation
strategies. Vast majority of the techniques proposed for noise
analysis are frequency-domain techniques that are applicable
to a particular circuit or a class of circuits. In time-domain,
stochastic differential equations are proposed in [3] to obtain
the time-varying covariance matrix of nonlinear circuits (e.g.
oscillators in [4], PLLs in [5], discrete- and continuous-time
circuits in [6]), and in [7], to obtain average and instantaneous
power spectral density (PSD). In [6], a numerical methods for
the efficient solution of stochastic differentials for noise
analysis is proposed.
In this paper, we extend a time-domain, non-stationary
stochastic noise analysis in [6] with a cyclo-stationary
stochastic process to include the treatment of sampled data
systems, and correspondingly, we derive the periodically timevarying spectral densities of such a process. In order to
characterize the fundamental limits of the sensing process and
post-processing interface circuit e.g. amplifier and A/D
converter, we treat the neural cell noise fluctuations as
homogeneous and inhomogeneous Markov chains and
interface electronics noise as a non-stationary stochastic
process. This approach provides key insight required to
address signal-to-noise ratio (SNR), response time, and
linearity of the physical electronic interface (i.e., saturation
level).
This research was supported in part by the European Union and the Dutch
government as part of the CATRENE program under Heterogeneous
INCEPTION project.
II.
STOCHASTIC NOISE ANALYSIS
A. Architectural Overiew of an Integrated Neural Interface
The signal quality in neural interface front-end, beside the
specifics of the electrode material and the electrode/tissue
interface, is limited by the nature of the bio-potential signal
and its biological background noise, dictating system resource
constraints (power, size, bandwidth, and thermal dissipation
i.e. to avoid tissue damage). The block diagram of a Mchannel neural recording system architecture is illustrated in
Figure 1. Due to the small amplitude of neural signals and the
high impedance of the electrode tissue interface, amplification
and low-pass filtering of the extracellular neural signals is
performed before the signals can be digitized. An inputreferred noise of an integrated front-end negative-feedback
amplifier needs to be smaller than those of electrode and
biological background noise. A successive approximation
register (SAR)-based A/D converter digitizes the amplified
neural signals with 10 bits of resolution. A low-power
monolithic digital signal processing (DSP) unit provides
additional filtering and executes a spike discrimination and
sorting algorithms. The relevant information is then
transmitted to an outside receiver through the transmitter or
used for K-channel stimulation in a closed-loop framework.
B. Noise Models
Neural cell noise model: In the Hodgkin and Huxley
framework, an neural channel’s configuration is determined
by the states of its constituent subunits, where each subunit
can be either in an open or closed state [8]. Adding a noise
term χx(V,t) (x=m,h, or n) to the deterministic ordinary
differential equation (ODE) of Hodgkin and Huxley is
consistent with the behavior of the Markov process for
channel gating [9]. Such process can be contracted to a
Langevin description (via a Fokker–Planck equation) and
expressed as delta-correlated noise processes Γneuron(t+τ,t)=
1/κ[αx(1-x)+βxx]δ(τ), where κ is the total number of neural
channels, and the transition rates αx(t) and βx(t) are
instantaneous functions of the membrane potential V(t).
Dirac’s delta function δ designates that the noise at different
times is uncorrelated and the variables m, h, and n represent
the aggregated fraction of open subunits of different types,
aggregated across the entire cell membrane.
#M
front-end neural interface
#M
recording
electrode
DSP
LNA
M
low noise
amplifier
#K
#K
#M
#M
M
back-end signal processing
M
band-pass
filter
n
M
programmable
gain amplifier
SAR A/D
converter
n
digital signal
processing system
K
D/A
converter
K
reconstruction stimulator
electrode
filter
Figure 1: Block diagram of a brain machine interface with front-end neural interface and back-end signal processing of an integrated neural implant.
Electrode-tissue interface noise model: The overall noise of
an electrode-tissue interface has contributions from the
tissue/bulk thermal noise, the electrode-electrolyte interface
noise, and the electronic noise. The most important types of
electrical noise sources (thermal, shot, and flicker noise) in
passive elements and integrated circuit devices have been
investigated extensively, and appropriate models derived [10]
as stationary and in [3] as nonstationary noise sources. We
adapt model descriptions as defined in [3], where thermal and
shot noise are expressed as Γthermal(t+τ,t)=2kTG(t)δ(τ) and
Γshot(t+τ,t)=qID(t)δ(τ), respectively, where k is Boltzmann’s
constant, T is the absolute temperature, G is the conductance,
q is the electron charge, and ID is the current through the
junction. These noise processes correspond to the current
noise sources, which are included in the models of the
integrated circuit devices. Tissue noise is modelled as the
thermal noise generated by the solution/spreading or
tissue/encapsulation resistance [11] and the electrode noise is
the thermal noise generated by the charge transfer resistor
[12]. The noise of the recording electronic circuits is mainly
determined by the thermal and flicker noise generated by the
input amplifier. Although the preamplifier can provide firstorder low-pass filtering, dedicated low-pass filters are used to
further minimize high-frequency noise. The cut-off frequency
of low-pass filters is set to fNeuron=10 kHz, where fNeuron is the
signal bandwidth of the action potential.
A/D converter noise model: Sampled data systems operate
on the series of discrete-time samples taken at the end of the
sampling period. Although the details of the processing during
each period result in nonstationary noise voltages and currents,
the same operation is performed each clock cycle, leading to
the same signal statistics each clock cycle. Consequently, such
stochastic process can be described as wide-sense
cyclostationary. The special case of a white noise input source
is of particular importance since the majority of the noise
sources can be traced back to white noise generated in circuit
components. For a white noise step input, the autocorrelation
is a delta function, where Sxo is the one-sided white noise
power spectral density (PSD) of the underlying noise process.
By using Parseval’s theorem, the variance of the output as a
function of the autocorrelation simplifies to Γ(t+τ,t)=
1/2Sxo(t)δ(τ) [13]. The one-sided noise PSD of the sampled
output can then be found from the sum of the filtered and
shifted two-sided input noise PSD Sx(f) [14]. Measurements of
the output codes for a dc input signal to the A/D converter can
be used to obtain an input-referred noise PSD estimate,
SADC(f). The noise of the input sampler and the converter
quantization noise add to the input-referred noise PSD to give
the total input noise PSD Stotal(f)=Ssample(f)+SADC(f)+Sq(f),
where Ssample(f)=(kT/Cs)/(fs/2) is the noise PSD from the input
sampler over the Nyquist range (0≤fNeuron≤fs/2) and Sq(f)=
(VLSB2/12)/(fs/2) is the A/D converter quantization noise.
C. Spectal Density and Time-Varying Variance
Consider description of the front-end neural interface with a
system of stochastic differential equations
F(r ' , r, t ) + B(r, t ) ⋅ χ = 0
(1)
where r is the vector of stochastic processes that represents
the state variables r(t) (e.g. node voltages), and B(r,t) is state
and time dependent modulation for the vector of noise
sources. χ is a vector of white Gaussian processes χ(t) i.e. the
derivative of the standard Wiener process, with the
autocorrelation function given by Γ(t+τ,t)=E[χ(t+τ)χ(t)]. Since
noise is treated as a perturbation, a system of nonlinear
stochastic differential equations in (1) is piecewise-linearized
with λ=[(r−r0)T,(χ−χ0)T]T; the partial results are than
combined to obtain the desired approximation. We will
interpret (1) as an Itô system of stochastic differential
equations. Such definition is consistent with the Markov
process for neural channel gating with continuous sample pats.
Using the multidimensional Itô formula, the variance matrix
K(t) of λ(t) is expressed in differential Lyapunov matrix form
as
dK (t ) / dt = E(t )K (t ) + K (t )E(t )T + B(t )B(t )T
(2)
where E(t)=F’(.) and Kij(t)=Ε[λi(t)λj(t)T], with the timevarying spectral, cross-spectral density matrix Sλ(t,f)=
F{K(t,τ)}. Correspondingly, the instantaneous periodically
time-varying spectral densities S(f) of the cyclostationary
stochastic process i.e. in A/D converters, can be expressed as
the derivative with respect to time of the expected noise PSD
dK ' (t ) / dt = E(t )K ' (t ) + K (t ) exp( jωt )
(3)
*
where the elements of the vector Ki’=E{λi(t)R(t,ω) } [15],
and R(t,ω) is the Fourier transform of a t+τ-segment of the
noise waveform. To solve the system in (3), we need to
calculate the time-varying transfer functions from the noise
source χ(t) to the components of r(t), as defined in (2). To
obtain a numerical solution, (2) has to be discretized in time
using a suitable scheme, such as any linear multi-step method,
or a Runge-Kutta method. If backward Euler is applied to (2),
the differential Lyapunov matrix equation is written in a
special form referred to as the continuous-time algebraic
Lyapunov matrix equation
E r K (t r ) + K (t r )ETr + B r BTr = 0
(4)
K(t) at time point tr is calculated by solving the system in
(4). In the circuit analysis, the matrix Er is, in general, not a
full-rank matrix; it may have zero rows and columns, and,
similarly, the node equation corresponding to this node will
not contain any time-derivatives. Consequently, we rewrite (4)
as a sparse linear matrix-vector system in standard form and
solve it with adjusted alternating direction method
K i = ( E rT − γ i I n )( E rT + γ i I n ) −1 K i −1 ( E r − γ i I n )( E r + γ i I n ) −1 (5)
- 2γ i ( E rT + γ i I n ) −1 Br BrT ( E r + γ i I n ) −1
for iterates i = 1,2,… This method generates a sequence of
matrices Ki, which converges fast towards the solution for
sparse matrices with small bandwidth [16], provided that the
iteration shift parameters γi are chosen (sub)optimally. Since
iteration starts with K0=0, Ki will have rank at most i×n,
where n is the number of vectors in Br. As a result, a sequence
of matrices Ki needs only to be represented by less than i×n
vectors.
40
20
0
-20
-40
-60
-80
0
50
100
150
200
250
Time (ms)
300
350
400
Figure 2: Nominal (without noise) voltage trace of neuron cell activity; the
complex spike burst is followed by a pause in spike activity.
60
Membrane potential [mV]
The proposed method allows for any deterministic neuron
model, whose neural conductance is described by a Markov
process, to be converted into an equivalent stochastic version
without involving any heuristics on the choice of the
parameters for extra noise sources. The covariance matrix is
periodic with the same period as either the input signal (e.g.,
translinear amplifier) or the clock (in sampled data systems).
Moreover, with the proposed method, it is also possible to
compute all the cross-correlations between any electrical
quantity and any stochastic source. This makes it possible for
the designer to determine the fundamental limit of the
system’s dynamic range, so that design efforts can be
addressed to the most critical section of the circuit and avoid
deteriorating power and chip area by overdesign. All the
experimental results are carried out on a single processor
Ubuntu Linux 9.10 system with Intel Core 2 Duo CPUs 2.66
GHz processor and 6 GB of memory. The proposed method
and all sparse techniques have been performed in a numerical
computing environment [17].
In time domain, a widespread approach for noise simulation
is Monte Carlo (MC) analysis. However, accurately
determining the noise requires a large number of simulations,
so consequently, the Monte Carlo based methods become cputime consuming in complex, multichannel neural interface.
The time series representation of an neuron signal at the
preamplifier’s input (Figure 2) are composed of a spike burst,
plus additive Gaussian white noise (Figure 3, grey area with
1000 randomly selected neural channel compartments and
black area with filtered out predicted bias from the estimated
variance σ2). Difference in standard deviation of the proposed
method vs. Monte Carlo analysis for 1000 iterations is within
Membrane potential (mV)
EXPERIMENTAL RESULTS
40
20
0
-20
-40
-60
-80
0
5
10
15
20
25
Time [ms]
Figure 3: Statistical voltage trace of neuron cell activity; grey area voltage traces from 1000 randomly selected neural channel compartments,
black area – expected voltage trace.
0
10
Normalized residual norm
III.
5%; cpu-time is reduced from 2 hours to 46 sec for 25 ms of
model time. Figure 4 illustrates the number of iteration steps.
After low- and high-pass filtering and amplification, the noisy
neural signal is sampled with SAR A/D converter. An
example of the time-domain noise estimation and noise power
spectral density at the output of the low-pass filter is
illustrated in Figure 5 and Figure 6, respectively. For
frequencies higher than ~10 kHz, capacitances at the interface
form the high-frequency pole and shape both the signal and
the noise spectrum; the noise is low-pass filtered to the
recording amplifier inputs. The interface’s input equivalent
noise voltage decreases as the gain across the amplifying
stages increase (Figure 7), e.g. the ratio of the square of the
signal power over its noise variance can be expressed as
SNR=FΣ2/(σ2neural+σ2electrode+Σi(ΠjGj-1)σ2amp,i), where FΣ is the
total signal power, σ2amp,i represents the variance of the noise
added by the ith amplification stage with gains Gj, σ2electrode is
the variance of the electrode and σ2neural is variance of the
biological neural noise. The observed SNR of the system also
increases as the system is isomorphically scaled up, which
suggests a fundamental trade-off between SNR and speed of
the system.
-5
10
-10
10
-15
10
-20
10
0
5
10
15
20
Iteration steps
25
30
35
Figure 4: Stopping criterion: maximal number of iteration steps.
IV.
-3
x 10
Amplitude [V]
5
0
-5
0
2
4
6
8
10
12
Time [ms]
14
16
18
20
Figure 5: Noise amplitude in time-domain at the output of the low-pass
filter; in comparison with 1000 Monte Carlo trials, the accuracy is within
3.2% with 21-fold cpu-time reduction.
-80
In this paper, we solve the set of linear time-varying
equations, including the noise content description to find the
steady state value of the time-varying variance matrix. The
variance matrix is periodic with the same period as either the
input signal (e.g., translinear amplifier) or the clock (in
sampled data systems). Moreover, we compute all the crosscorrelations between any electrical quantity and any stochastic
source and compute noise spectral density. In a neural
interfaces with large biological noise, this makes it
conceivable to determine the limit of the system’s dynamic
range, and avoid deteriorating power and chip area by
overdesign.
REFERENCES
rms
Magnitude [dBV2 /Hz]
-100
[1]
-120
-140
-160
[2]
-180
0
10
1
[3]
2
10
Frequency [kHz]
10
Figure 6: Noise PSD at the output of the low-pass filter.
0
[4]
dB
-50
[5]
-100
[6]
-150
1
10
2
10
3
10
Frequency [Hz]
4
10
10
5
[7]
Figure 7: Spectral signature of SAR A/D converter-two tone test; black
area – spectral content with nominal gain, grey area – spectra with 20% gain
reduction, equivalent to 4 LSB loss in the dynamic range.
Design
CONCLUSION
Area
[mm2]
0.096
0.052
0.066
0.058
0.036
0.074
0.042
0.424
SNR (100Hz-10kHz) [dB]/channel
WCD
MC
SNA
slow, nom, fast [dB]
[dB]
rel.
58.5, 60.9, 62.5
58.9
4.3
57.1, 58.9, 59.6
58.1
3.2
56.3, 58.2, 59.7
56.9
4.1
59.3, 60.7, 61.4
60.2
5.6
56.7, 58.7, 59.4
57.4
2.9
57.9, 60.7, 61.6
59.3
6.2
62.5, 64.6, 65.6
63.7
4.3
55.4, 57.3, 58.3
56.9
6.4
4.6
LNA
LPF
HPF
PGA
SARcomp
RDAC
SARlogic
Total
Average
TABLE I– SUMMARY OF THE ALGORITHM PERFORMANCE
t(cpu)
rel.
35.4
21.7
23.4
43.4
17.2
24.6
32.3
38.8
29.6
In Table 1, SNR/channel estimated with worst-case analysis
(WCD) and 1000 Monte Carlo (MC) iterations is compared
across the neural interface circuits with the proposed
stochastic noise analysis (SNA). Our approach allows large
cpu-time reduction, ranging from 17-fold to 43-fold, with 29fold on average. Difference in standard deviation of SNA vs.
Monte Carlo analysis is less than 4.7% on average.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
F. Chen, A.P. Chandrakasan, V.M. Stojanovic, “Design and analysis of a
hardware-efficient compressed sensing architecture for data compression
in wireless sensors,” IEEE Journal of Solid-State Circuits, vol. 47, no. 3,
pp. 744-756, 2012.
R. Harrison, “The design of integrated circuits to observe brain activity,”
Proceedings of IEEE, vol. 96, no. 7, pp. 1203-1216, 2008.
A. Demir, E. Liu, A. Sangiovanni-Vincentelli, “Time-domain nonMonte Carlo noise simulation for nonlinear dynamic circuits with
arbitrary excitations,” Proceedings of IEEE International Conference on
Computer Aided Design, pp. 598-603, 1994.
A. Demir, A. Mehrotra, J. Roychowdhury, “Phase noise in oscillators: A
unifying theory and numerical methods for characterization,” IEEE
Transactions on Circuits and Systems-I, vol. 47, no. 5, pp. 655-674,
2000.
A. Mehrotra, “Noise analysis of phase locked loops,” IEEE Transactions
on Circuits and Systems-I, vol. 49, no. 9, pp. 1309-1316, 2002.
A. Zjajo, et al., “Stochastic analysis of deep-submicrometer CMOS
process for reliable circuits designs,” IEEE Transactions on Circuits and
Systems-I, vol. 58, no. 1, pp. 164-175, 2011.
V. Vasudevan, “A time-domain technique for computation of noisespectral density in linear and nonlinear time-varying circuits,” IEEE
Transactions on Circuits and Systems-I, vol. 51, no.2, pp. 422-433,
2004.
A. Hodgkin, A. Huxley, “A quantitative description of membrane
current and its application to conduction and excitation in nerve,”
Journal of Physiology, vol. 117, pp. 500-544, 1952.
R.F. Fox, Y.-N. Lu, “Emergent collective behavior in large numbers of
globally coupled independently stochastic ion channels,“ Physics Review
E., vol. 49, pp. 3421-3431, 1994.
P.R. Gray, R.G. Meyer, Analysis and Design of Analog Integrated
Circuits, New York: Wiley, 1984.
A.C. West, J. Newman, “Current distributions on recessed electrodes”,
Journal of the Electrochemical Society, vol. 138, no. 6, pp. 1620-1625,
1991.
Z. Yang, Q. Zhao, E. Keefer, W. Liu, “Noise characterization, modeling,
and reduction for in vivo neural recording,” Advances in Neural
Information Processing Systems, pp 2160-2168, 2010.
J.H. Fischer, “Noise sources and calculation techniques for switched
capacitor filters,” IEEE Journal of Solid-State Circuits, vol. 17, no. 4,
pp. 742-752, 1982.
T. Sepke, P. Holloway, C.G. Sodini, H.-S. Lee, “Noise analysis for
comparator-based circuits,” IEEE Transactions on Circuits and SystemsI, vol. 56, no. 3, pp. 541-553, 2009.
L. Arnold, Stochastic Differential Equations: Theory and Application,
New York: Wiley, 1974.
E. Wachspress, “The ADI minimax problem for complex spectra”,
Applied Mathematical Letters, vol. 1, pp. 311-314, 1988.
MatLab, http://www.mathworks.com/.